#6937399

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30560	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,30,35,31,36,27,32)(28,39,37,47,41,50,44,33)(29,34,43,49,42,48,38,45)(40,52,51,55,53,56,54,46);(1,2)(3,6)(4,5)(7,8)(10,14)(12,19)(15,20)(18,23)(25,27,31,30)(26,32,36,35)(28,37,41,44)(29,38,42,43)(33,39,47,50)(34,45,48,49)(40,51,53,54)(46,52,55,56);(2,5)(3,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,31,41)(26,33,36,47)(27,37,30,44)(29,40,42,53)(32,39,35,50)(34,46,48,55)(38,51,43,54)(45,52,49,56);(1,3,4,7)(2,6,5,8)(10,14)(11,17)(16,21)(18,23)(25,29,31,42)(26,34,36,48)(27,38,30,43)(28,40,41,53)(32,45,35,49)(33,46,47,55)(37,51,44,54)(39,52,50,56);(2,5)(6,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(29,42)(34,48)(38,43)(40,53)(45,49)(46,55)(51,54)(52,56)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844595);